L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + (1.51 − 1.64i)5-s + 0.999i·6-s + (0.0701 + 0.0404i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (−2.18 − 0.483i)10-s − 3.93·11-s + (0.866 − 0.499i)12-s + (1.78 − 3.09i)13-s − 0.0809i·14-s + (−2.13 + 0.672i)15-s + (−0.5 − 0.866i)16-s + (0.563 + 0.975i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.675 − 0.737i)5-s + 0.408i·6-s + (0.0264 + 0.0152i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−0.690 − 0.152i)10-s − 1.18·11-s + (0.249 − 0.144i)12-s + (0.495 − 0.857i)13-s − 0.0216i·14-s + (−0.550 + 0.173i)15-s + (−0.125 − 0.216i)16-s + (0.136 + 0.236i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6589530661\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6589530661\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (-1.51 + 1.64i)T \) |
| 37 | \( 1 + (-1.49 + 5.89i)T \) |
good | 7 | \( 1 + (-0.0701 - 0.0404i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 3.93T + 11T^{2} \) |
| 13 | \( 1 + (-1.78 + 3.09i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.563 - 0.975i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.70 + 0.983i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 2.93T + 23T^{2} \) |
| 29 | \( 1 - 2.65iT - 29T^{2} \) |
| 31 | \( 1 + 6.08iT - 31T^{2} \) |
| 41 | \( 1 + (3.72 - 6.44i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 5.69T + 43T^{2} \) |
| 47 | \( 1 + 10.7iT - 47T^{2} \) |
| 53 | \( 1 + (0.118 - 0.0685i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.46 - 1.42i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-8.34 - 4.81i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.06 + 1.76i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.49 - 6.04i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 5.98iT - 73T^{2} \) |
| 79 | \( 1 + (3.95 + 2.28i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (10.6 - 6.14i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (11.3 - 6.52i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 0.882T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.669240386782114820316847371745, −8.415612873515036644865570631348, −8.149747026043515876575859075978, −6.93885013752410359233181765952, −5.75115093676467175977884605239, −5.26636179046074079881651969375, −4.12218649027838157703902461465, −2.73223301541423961961796571063, −1.68283845130453047618243333713, −0.33988567549595269798953887570,
1.72508533571879713419748778384, 3.07447450750449067783727586892, 4.43981983333196131634250692862, 5.37633129084395739368676953208, 6.16082528544067061140075002525, 6.79655778090771193649082586923, 7.72519265489444410130463716533, 8.627242985855074781512459657533, 9.598010261680794825881061629196, 10.20568823345775622920905550290